Printing a mask with maximum possible process window through adjustment of the source distribution

ABSTRACT

A lithographic mask is illuminated with light from different directions such that intensities of a plurality of incident beams of light provide a largest possible integrated process window defined in terms of an allowed range for defining shapes. Constrained sets of intensity parameters are imposed. A first set of intensity parameters represents maximum possible intensities that can be permitted for overexposed tolerance positions. A second set of intensity parameters represents minimum possible intensities that can be permitted for underexposed tolerance positions. Optimum source intensities of incident beams are defined using a linear program and constraints. The optimum source intensities maximize an integrated range of dose and focal variations without causing printed shapes to depart from the allowed range. Apparatus are detailed and variations are described.

CROSS REFERENCE TO RELATED APPLICATION

This application claims priority to U.S. patent application Ser. No.11/377,957, filed on Mar. 16, 2006 now U.S. Pat. No. 7,363,611; andthrough that application further claims priority to U.S. patentapplication Ser. No. 10/727,901, filed on Dec. 4, 2003 and issued onJun. 6, 2006 as U.S. Pat. No. 7,057,709. The former application is adivisional of the latter application which is now issued. Thisapplication is also a divisional of that latter application which is nowissued.

TECHNICAL FIELD OF THE INVENTION

This invention relates to manufacturing processes that project highresolution images by illuminating a mask, and in particular to methodsfor maximizing the manufacturing process window by adjusting theillumination on the mask.

BACKGROUND OF THE INVENTION

Optical lithography has been widely used in the formation of structuresincluded in integrated circuit (IC) chips. Reductions in the size ofstructures within IC chips have increased the demands placed uponoptical lithography systems.

Many methods have been developed to compensate for the image degradationthat occurs when the resolution of optical lithography systemsapproaches the critical dimensions (CD's) of desired lithographicpatterns. Critical dimension (CD) refers to the feature size and spacingbetween features and feature repeats (pitch) that are required by thedesign specifications, and that are critical for the proper functioningof the devices on a chip. When the critical dimensions (CDs) of adesired IC pattern approach the resolution of a lithographic system(defined as the smallest dimensions that can be reliably printed by thesystem), image distortions become a significant problem. Currently, thelimited resolution of lithography tools poses a challenge toimprovements in IC manufacture. The importance of resolution continuesto increase as critical dimensions continue to decrease. In order tomake the manufacture of future IC products feasible, lithography toolswill be required to achieve adequate image fidelity when the ratio ofthe minimum CD to resolution of the lithographic system is very low.

As an introduction, the resolution ρ (rho) of an optical lithographysystem can be described by the equation:

$\rho = \frac{k\;\lambda}{NA}$where ρ is the is the minimum feature size that can be lithographicallyprinted, NA (numerical aperture) is a measure of the amount of lightthat can be collected by the lens, λ (lambda) is the wavelength of thelight source, and k is a factor unique to a given system. One canunderstand that the resolution ρ is proportional to the wavelength ofthe light source, and that the image fidelity is improved as diffractedlight is collected by the lens over a wider range of directions (i.e.,as NA increases). Although a larger NA permits smaller features to beprinted, in practice NA is limited by depth-of-focus requirements, bypolarization and thin-film effects, by the finite refractive index ofthe medium (usually air) at the lens exit, and by limitations in lensdesign. The factor k accounts for aspects of the lithographic processother than wavelength (λ) or numerical aperture (NA), such as resistproperties or the use of enhanced masks. In the prior art, typicalk-factor values range from about 0.4 to about 0.7. Because oflimitations in reducing wavelength λ or increasing numerical apertureNA, the manufacture of smaller IC features (having smaller CD's) willrequire reducing the k-factor to, for example, the range 0.3-0.4 orsmaller, in order to improve the resolution of the lithographicprocesses.

Components of one embodiment of a projection lithographic system 520 areillustrated in FIG. 1. In FIG. 1, an illumination source 524 providesradiation that illuminates a mask 526, also known as a reticle 526. Theillumination source 524 is typically controlled by an illuminationcontroller 522. The terms “mask” and “reticle” may be usedinterchangeably. Typically, the reticle 526 includes features that actto diffract the illuminating radiation through a lens 530 which projectsan image onto an image plane 532, for example, a semiconductor wafer 550as a substrate 550. The directional extent of radiation transmitted fromthe reticle 526 to the lens 530 may be controlled by a pupil 401. Theillumination source 524 may be capable of controlling various sourceparameters such as direction and intensity. The wafer 550 typicallyincludes a photoactive material (known as a resist). When the resist isexposed to the projected image, the developed features in the resistclosely conform to the desired pattern of features required for thedesired IC circuit and devices.

The pattern of features on the reticle 526 acts as a diffractingstructure analogous to a diffraction grating which transmits radiationpatterns. These radiation patterns can be conveniently described interms of a Fourier transform in space based on spacing of the featuresof the diffraction grating (or reticle 526). The Fourier components ofdiffracted energy associated with the spatial frequencies of thediffracting structure are known in the art as diffracted orders. Forexample, the zeroth order is associated with the DC component of themask Fourier transform, while higher Fourier orders arise from modulatedpatterns in the mask, and are related to the wavelength of theilluminating radiation and inversely related to the spacing (known aspitch) between repeating diffracting features. When the pitch offeatures is smaller, the angle of diffraction is larger, so that somehigher diffracted orders will be diffracted at angles larger than thenumerical aperture (NA) of the lens 530. The system 520 may includeother apparatus as appropriate, such as, for example, optical filters.

FIG. 5, discussed further herein in the context of an exemplaryembodiment, depicts aspects of pixels 1-40 produced by the source 524.FIG. 5 provides a view of the pupil 401, wherein an illumination pupil402 is also shown, the illumination pupil 402 being that portion of thepupil 401 that is accessible to the illumination source 524. Theillumination pupil 402 is typically a central region of the pupil 401,and may be up to about 0.9 NA.

There is increasing interest in methods to optimize the illuminationdistributions used in photolithography to provide for these smallstructures. Exemplary U.S. patents include U.S. Pat. No. 5,680,588,“Method and System for Optimizing Illumination in an OpticalPhotolithography Projection Imaging System,” issued to Gortych et al,Oct. 21, 1997; and U.S. Pat. No. 6,563,566, “System and Method forPrinting Semiconductor Patterns Using an Optimized Illumination andReticle” issued to Rosenbluth et al., May 13, 2003.

Other publications in this area include “Illuminator Design For ThePrinting Of Regular Contact Patterns,” M. Burkhardt, A. Yen, C. Progler,and G. Wells, Microelectronic Engineering 41-42, 1998, p. 91; “TheCustomized Illumination Aperture Filter for Low k1 PhotolithographyProcess,” T.-S. Gau, R.-G. Liu, C.-K. Chen, C.-M. Lai, F.-J. Liang, andC. C. Hsia, SPIE v.4000—Optical Microlithography XIII, 2000, p. 271;“Optimum Mask and Source Patterns to Print a Given Shape,” A. E.Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, A. Molless, R. N.Singh, and A. K. Wong, Journal of Microlithography, Microfabrication,and Microsystems, Vol. 1, No. 1, 2002, p. 13; and, “IlluminationOptimization of Periodic Patterns for Maximum Process Window,” R. Socha,M. Eurlings, F. Nowak, and J. Finders, Microelectronic Engineering61-62, 2002, p. 57.

The publication entitled “Optimum Mask and Source Patterns to Print aGiven Shape,” as well as U.S. Pat. Nos. 5,680,588 and 6,563,566,describe methods for mathematically optimizing the illuminationdistribution (“source”) to obtain the sharpest possible image in focus.However, the publication shows that the sharpness of the focused imagedoes not always correlate with the overall lithographic quality, since asmall sacrifice in exposure latitude at best-focus can yield a largeincrease in depth-of-focus (DOF).

Such complexities in lithographic tradeoffs have long been appreciatedoutside of the context of numerical optimization. The so-calledED-window (exposure defocus) analysis is a convenient way to assesslithographic quality that takes into account both exposure latitude anddepth-of-focus. The integrated area of the ED-window is a usefulsingle-parameter metric for assessing overall image quality. Moreinformation in this regard is provided in the publication“Level-Specific Lithography Optimization for 1-Gb DRAM,” A. K. Wong, R.Ferguson, S. Mansfield, A. Molless, D. Samuels, R. Schuster, and A.Thomas, IEEE Transactions on Semiconductor Manufacturing 13, No. 1,February 2000, p. 76. It is noted that the integrated ED-window, alsoreferred to as the “integrated process window,” is the integral withrespect to focal range Δz of the fractional exposure latitude obtainedthroughout each focal range.

Solving for a maximal integrated ED-window is problematic. This isbecause the ED metric involves factors that are conditional, non-linear,and non-differentiable. For example, the exposure latitude attained in agiven focal range is locally unaffected by all critical distances (CD)in the pattern, except for a particular CD where the exposuresensitivity is the worst for the pattern. That is, the overallperformance of the lithographic process is limited to the performancefor the weakest feature. Thus, the ED metric reflects a requirement thatall features must be printed within a specified tolerance. Similarly,the exposure latitude attained in the extreme defocused plane of a givenfocal range is not counted as the final exposure latitude applying inthat range. Rather, the quantity that is restrictive is the worst caseexposure latitude attained in all of the focal planes within the range.Also, even though image intensity (the reciprocal of exposure) is linearin the source intensity, it is usually only the fractional exposurelatitude (a nonlinear quantity) that is considered important, since theabsolute dose level can typically be adjusted by changing the exposuretime. Finally, the result of integration over the focal range that isused to obtain the final figure of merit (i.e., the integrated EDwindow) must be truncated at the depth-of-focus range where the exposurelatitude first drops to zero. This variable truncation representsanother non-linear quantity.

These complications limit the reliability of heuristic methods foroptimizing lithographic sources over extended focal ranges. Prior artheuristic methods are described in U.S. Pat. Nos. 5,680,588 and6,563,566, as well as in the publication entitled “IlluminationOptimization of Periodic Patterns for Maximum Process Window.” Some ofthese methods attempt to optimize the image with respect to focus andexposure by applying the focused-image algorithm in multiple focalplanes. One may note that an image which has been optimized acrossmultiple focal planes may still exhibit variation in CD through thedepth-of-focus, even though the image has been made sharp in each focalplane. Within the context of the teachings for this prior art, onemethod of addressing this problem is through the addition of CDconstraints across the depth-of-focus. However, such equality or bandconstraints would predictably require adjustment on an ad-hoc oriteration-by-iteration basis, since the depth-of-focus and exposurelatitude tradeoff is highly variable from pattern to pattern, andtherefore difficult to predict.

Another prior art heuristic algorithm is introduced in the publicationentitled “Illumination Optimization of Periodic Patterns for MaximumProcess Window.” In this algorithm, the intensity given to each sourcepoint (i.e., each illumination direction) is in one embodimentcalculated by forming products of two weighting factors, and thensumming these products over all pairs of amplitude diffraction ordersthat the lens 530 collects from the source direction in question. Afirst of two weighting factors used is (the absolute value of) theintensity spatial frequency produced by the pair of amplitudes. Thus,weighting provides for preferentially augmenting high spatialfrequencies in order to sharpen the image. The second weighting factoris obtained by first calculating the normalized derivative with respectto focus of the particular two-order intensity in question. Theweighting factor is then calculated by subtracting from 1 the ratio ofthe focal slope to the maximum focal slope encountered amongst allsource points and image frequencies (so that this weighting factorranges from 1 to 0). Variations of the method are provided, all of whichinvolve similar weighting heuristics. These heuristics are plausiblydesigned and have been shown to perform acceptably in certain examples.

However, and as with the other methods discussed above, images improvedby heuristic methods are not guaranteed to meet standard lithographiccriteria for optimal behavior through the depth-of-focus. For example,lithographic performance is customarily considered to be limited by thequality of the least robust printed feature or CD (i.e., the relevantcriterion is whether or not every feature is printed within tolerance;if one feature fails, the successful performance of all other featuresis effectively irrelevant).

Unfortunately, a filtering algorithm that emphasizes features withminimal focus and exposure sensitivity can underemphasize the weakest(and therefore most critical) features in the image. Althoughadjustments could be introduced on an ad-hoc basis to deal with suchlimitations, these typically do not directly address the problem. Withheuristic algorithms, one must usually expect such difficulties willarise, and performance will typically become less reliable when theheuristic adjustments are made more abstract (i.e., when the adjustmentsdo not directly embody the features of the problem which must besolved). On the other hand, standard general-purpose methods foroptimization, though non-heuristic, are based on refinement of aninitial starting design, and so are limited by the quality of thisinitial design (i.e., the solutions provided are usually no better thanthe particular local optimum associated with the design used to startoptimization). This local optimum is not, in general, the desired globaloptimum. Some general-purpose optimization methods, for example geneticalgorithms, do attempt to search beyond the initial local optimum, butthere is no guarantee that they will find the global optimum. This isparticularly true when more than a few variables are involved, since thepotential search space becomes immense. Heuristic methods in some senseconsider the properties of the entire search space, but they do so in anuncertain way, and they also do not guarantee that the globally optimumsolution is found.

Despite some inherent weakness, heuristic algorithms can often improvethe quality of a lithographic image. However, these algorithms leaveroom for improvement. For example, one would usually expect sourceweightings that are chosen by heuristic methods to produce appreciableCD errors; correction of these errors using known techniques (e.g. usingan optical proximity correction (OPC) program) will in turn change thebehavior of the image, and there is no guarantee that iterated cycles ofheuristic source optimization and OPC adjustment will converge to anoptimal solution.

Although heuristic algorithms can deliver better lithographic imagesthan earlier techniques, the required adjustments to same are notconducive to routine use. What is needed is a method for illuminatingthe mask 526 with a source 524 that may be used to optimize the imageaccording to an accepted lithographic criterion, such as maximization ofthe integrated process window through focus.

SUMMARY OF THE INVENTION

The foregoing and other problems are overcome by methods and apparatusin accordance with embodiments of this invention. Disclosed herein aretechniques to produce a source pattern for illumination of alithographic mask in such a way that the maximum possible integratedprocess window is obtained.

One aspect of the teachings herein includes a mask illumination method,which calls for illuminating a lithographic mask with a source of lightfrom different directions such that intensities of a plurality ofincident beams of light provide a largest possible integrated processwindow defined in terms of an allowed range for defining shapes, byimposing, through application of at least one set of constraints, afirst set of intensity parameters for representing maximum possibleintensities that can be permitted for overexposed tolerance positionsand a second set of intensity parameters for representing minimumpossible intensities that can be permitted for underexposed tolerancepositions; defining, for each of a plurality of different focal ranges,at least one parameter for each of the first set and the second set;and, determining optimum source intensities using a linear program andconstraints that include at least said one set of constraints, where thedetermined optimum source intensities maximize an integrated range ofdose and focal variations without causing printed shapes to depart fromthe allowed range.

Another aspect of the teachings herein includes a method to conform aprojected pattern to a range of shapes within a maximized total range ofdefocus positions and exposure doses, by illuminating a set of maskpatterns from a plurality of source directions each having adjustableintensity, projecting an image of the mask patterns from each of thesource directions onto an adjustable total range of defocused planes,establishing proportionalities between the adjustable source intensitiesand the projected intensities along the darkest boundaries of the rangeof shapes, establishing proportionalities between the adjustable sourceintensities and the projected intensities along the brightest boundariesof the range of shapes, constraining a first limiting intensityparameter in each of the defocused planes to the maximum projectedintensity along the darkest boundaries of the range of shapes,constraining a second limiting intensity parameter in each of thedefocused planes to the minimum projected intensity along the brightestboundaries of the range of shapes, constraining the first limitingintensity parameter in each particular defocused plane to the maximum ofthe maximum intensities within the range of defocus positions bounded bythe particular defocused plane, constraining the second limitingintensity parameter in each particular defocused plane to the minimum ofthe minimum intensities within the range of defocus positions bounded bythe particular defocused plane, determining the values of the adjustablesource intensities and the total range of defocus planes which maximizethe difference between the sum of the maxima of maximum intensities andthe sum of the minima of minimum intensities, and adjusting theadjustable source intensities to the difference-maximizing values.

A further aspect of the teachings herein includes a method to obtain anillumination source solution for illuminating at least one mask forprinting a pattern defined by the at least one mask, by supplying maskshapes and an allowable range of printed shapes; determining samplepoints; using a source pixelation, determining proportionalities betweensource intensities and sample point intensities, for individual ones ofa plurality of focal planes; determining a maximum allowable intensityat dark sample points, and a minimum allowable intensity at brightsample points; constraining a first intensity parameter in each focalplane to a maximum intensity among sample points at dark boundaries ofthe shape range; constraining a second intensity parameter in each focalplane to a minimum intensity among sample points at bright boundaries ofthe shape range; constraining the first intensity parameter in eachfocal plane to a maximum intensity bound within a truncated focal range;constraining the second intensity parameter in each focal plane to aminimum intensity bound within the truncated focal range; and executinga focal range loop, that includes selecting an initial defocus limit;maximizing a difference between a sum of the bounded maximum intensitiesand a sum of the bounded minimum intensities; and iterating byincreasing the defocus limit so long as the constraints are met,otherwise terminating the focal range loop and outputting a result thatprovides a maximum difference between the sum of the maximum intensitiesand the sum of the minimum intensities.

Also taught herein is a system for illuminating a mask, that includesapparatus for illuminating a photolithographic mask with light fromdifferent directions such that intensities of a plurality of incidentbeams provide a largest possible integrated process window defined interms of an allowed range for defining shapes, the illuminatingapparatus including further apparatus for imposing, through applicationof at least one set of constraints, a first set of intensity parametersfor representing maximum possible intensities that can be permitted foroverexposed tolerance positions and a second set of intensity parametersfor representing minimum possible intensities that can be permitted forunderexposed tolerance positions; apparatus for defining, for each of aplurality of different focal ranges, at least one parameter for each ofthe first set and the second set; and apparatus for determining optimumsource intensities using a linear program and constraints that includeat least said one set of constraints, where the determined optimumsource intensities maximize an integrated range of tolerable dose andfocal variations without causing printed shapes to depart from theallowed range.

Another aspect of the teachings herein includes a method for minimizinga loss in an integrated process window, the loss resulting fromperturbations in a plurality of masks, each mask containing features forprinting a common shape, where the perturbations span the range oferrors that must realistically be expected when the features arefabricated on a typical mask using a practical mask-making process, themethod includes illuminating each mask with a source of light fromdifferent directions according to a common window, wherein illuminationis such that intensities of incident beams of light provide a largestpossible integrated process window for printing the common shape withinthe allowed range using every perturbed mask, with the same exposure andfocal range used for each mask, the integrated process window defined interms of an allowed range for defining the common shape, wherein thelargest possible integrated process window comprises an integrated rangeof dose and focal variations for printing the common shape withoutcausing the printed common shape to depart from the allowed range.

Also disclosed are teachings for a method to provide an empirical yieldimprovement in at least one printed feature produced by a lithographicsystem by, through application of at least one set of constraints,imposing a first set of intensity parameters for representing maximumpossible intensities that can be permitted for overexposed tolerancepositions and a second set of intensity parameters for representingminimum possible intensities that can be permitted for underexposedtolerance positions; defining, for each of a plurality of differentfocal ranges, at least one parameter for each of the first set and thesecond set; determining optimum source intensities using a linearprogram and constraints that include at least the one set ofconstraints; specifying a bias that provides for an offset from acritical dimension of a design specification for each printed feature;and, limiting changes in the source intensities to introduce the bias.

Aspects of methods disclosed herein may be implemented by computerexecutable instructions stored on computer readable media.

BRIEF DESCRIPTION OF THE DRAWINGS

The above set forth and other features of the invention are made moreapparent in the ensuing Detailed Description of the Invention when readin conjunction with the attached Drawings, wherein:

FIG. 1 illustrates basic components of a projection lithographic systemas known in the art;

FIGS. 2A-C, collectively referred to as FIG. 2, show an example of a setof lithographic target features which, when printed, have shapes fallingwithin a certain prescribed range;

FIGS. 3A-C, collectively referred to as FIG. 3, show an exemplary set ofmasks that correspond to the shapes appearing in FIG. 2;

FIGS. 4A-C, collectively referred to as FIG. 4, show an exemplary set ofimage sample points for testing shapes such as those appearing in FIG.2;

FIG. 5 shows aspects of the pixelation of the source, as would beapplied to the example in FIG. 2;

FIG. 6 shows an improved pixelation of the source, for the FIG. 2example;

FIG. 7 depicts the focused image produced by a particular source pixel,in the FIG. 5 example. This represents a trial image that would beobtained if the source pixel were given unit intensity;

FIG. 8 depicts the defocused trial image produced by the particularsource pixel of FIG. 7;

FIG. 9 depicts the focused trial image produced by another source pixelin the FIG. 5 example;

FIG. 10 depicts the defocused trial image produced by the particularsource pixel of FIG. 9;

FIG. 11 provides a flowchart for maximizing the integrated processwindow;

FIG. 12 provides a flowchart with further steps for maximizing theintegrated process window;

FIG. 13 depicts an illumination distribution that maximizes theintegrated process window for the example of FIG. 2; and,

FIG. 14 depicts an embodiment of a system for maximizing the integratedprocess window.

DETAILED DESCRIPTION OF THE INVENTION

Disclosed herein are methods and apparatus for maximizing the processwindow through adjustment of the source distribution.

As used herein, the process window is defined as a set of limits ortolerances for the various imaging parameters. The process window takesinto account the extremes of the range of lengths and widths that areallowed for the shapes, and the range of dose and focus variations whichprint all shapes within the allowed extremes. In many cases, one set ofshape limits corresponds to underexposed print conditions, and anotherset to overexposed conditions. Sample points may be used to delineatethe limits of allowable shapes according to two-dimensional criteria.

An optimum source pattern is expressed as a list of optimum values forthe intensity of light that should be input to the mask from eachpossible illuminating direction. The possible illuminating directions(also referred to as “source directions”) are defined using a griddingof direction space, which may also be referred to as a pixelation ofdirection space. Thus, the “source direction” may actually refer to asmall range of directions that are contained within one pixel containedin the source grid.

Once the source pixelation has been determined, proportionalities aredetermined between the intensity of each source direction and theresulting image intensity. This is performed for each of a number ofpossible focal planes. The intensities at image sample points areconsidered.

The source intensities are then transformed into ratios, defined asfractions of the total source intensity. A first set of limitingintensity parameters is produced. The first set represents the maximumintensity that arises at sample points on the relatively darkerboundaries of allowed shapes, in each of a number of focal planes. Thismaximum intensity determines the maximum exposure dose that can betolerated without producing an overexposed condition on the wafer. Thesefirst parameters are not actually set equal to the maximum intensities(as determined by the source intensity proportionalities). Instead,these limiting intensity parameters are constrained to be no smallerthan any of the source-determined intensities. Such inequalityconstraints (that are nominally less specific than equality constraints)allow the optimum source solution to be determined, as the inequalityconstraints are ultimately made to automatically yield equality.Similarly, a second set of limiting intensity parameters are preferablyconstrained to be no larger than the source determined intensities forunderexposed sample points. One parameter of each set is defined foreach of a number of different focal ranges. Other constraints aredefined, for example so as to require bright regions of the image toprint bright, and dark regions to print dark. Additional constraints maylimit the intensity that can be provided by a given source directionaccording to the brightness limitations of the exposure system.

An optimum set of source intensities can then be determined from theseconstraints, where the optimum source intensities are those thatmaximize the integrated range of intensity dose and focal variationsthat can be tolerated without causing the printed shapes to depart fromthe allowed range of shapes.

The teachings herein make use of a photolithography system wherein theillumination pupil 402 (see, for example, FIG. 6) is divided into pixels1-40 (referred to as “pixelated” or as “pixelation” as appropriate). Asdiscussed further below, the use of forty pixels, or pixels in thespecific position of pixels 1-40 as in FIG. 6, is only illustrative oftechniques involving the example depicted in FIG. 2. Accordingly, theexample provided in FIG. 2, and the ensuing discussion thereof, is notlimiting of the teachings herein.

Pixelation of the illumination pupil 402 is used to provide fordetermination of the optimum illumination distribution. The illuminationpupil 402 is that portion of the pupil 401 that is accessible to theillumination source. The illumination pupil 402 is typically about 0.9times the numerical aperture of the lens 530 (0.9 NA). Aspects ofoptimizing illumination in terms of pixelation are disclosed in U.S.Pat. No. 5,680,588, the disclosure of which is incorporated by referenceherein in its entirety.

The pixelation may use a programmable mirror array with very highpixel-density. In such a configuration, the pixels 1-40 that areemployed during optimization can be formed as groups of fine physicalpixels in the mirror array. Another example of pixelation involvesunfolding of the illuminating angular space that is defined by multiplefold reflections arising in a light tunnel homogenizer.

The pixelated image does not have to be densely packed, nor does it haveto be based on a Cartesian subdivision of the pupil. Individual pixels1-40 do not have to be contiguous. Preferably, the individual pixels1-40 represent source elements whose intensity can be individuallyadjusted. The algorithm disclosed herein can use virtually any set ofsource elements. Preferably, each source element may be represented by aset of illumination directions, with bilateral symmetry about horizontaland vertical planes, to avoid image skew through focus. The methoddisclosed herein has at least one advantage of being adaptable towhatever pixelation techniques a given exposure system implements.However, for purposes of the disclosure herein, a pixelation method thatprovides certain computational advantages is discussed. One skilled inthe art will recognize that the pixelation techniques discussed hereinare not limiting of these teachings, and therefore are merelyillustrative.

Once the illumination pixelation is defined, the optimum illuminationsource can be expressed formally as an unknown vector set {right arrowover (s)}. That is, {right arrow over (s)} is initially a list ofunknown optimum intensity values for each of the source elements. In theimage plane the intensity at a given point can then be written as a sumof contributions from each unknown source element. That is, theintensity at a given point in the image plane can be expressed as {rightarrow over (s)}·{right arrow over (I)}. Note that the intensitycontributions from different elements add incoherently, and that theintensities {right arrow over (I)} provided by the different elementscan be calculated using known methods, such as those presented in thetext “Principles of Optics” M. Born and E. Wolf, 5^(th) Ed., PergammonPress, 1975. For convenience, these intensities are referred to as“trial intensities.” Thus, the exposure (the reciprocal of intensity) ata given sample point in the image plane (in this case the i^(th) samplepoint) is expressed as:

$\begin{matrix}{E_{i} = {\frac{1}{\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{i}\left( {x,z} \right)}}.}} & \left( {{Eq}.\mspace{14mu} 1} \right)\end{matrix}$

A desired image shape is specified using sample points that map out thenominally bright and dark regions of the image, and the boundaries ofthe allowable range of shapes for the printed features. In oneembodiment, the shape is specified by a preliminary user-input step. Inanother embodiment, the sample points are automatically produced along agrid in the image.

A notational convention is used herein for distinguishing between brightand dark sample points where the subscript index i is replaced by eitherthe subscript index u for representing bright sample points, or thesubscript index v for representing for dark sample points.

Additional sample points are chosen near key target edges. That is, edgesample points are chosen at or near the desired edge positions of CDs inthe image. Edge sample points are denoted herein by the index subscriptr. For purposes of discussion, the CD slice for the r^(th) edge isconsidered to cut the aerial image along a coordinate x, independent ofwhether the orientation of the feature edge is actually horizontal,vertical, or non-Manhattan. The CD slice can thus be regarded as a plotof intensity versus x in a direction through the sample point that islocally perpendicular to the feature edge (which could in general becurved, e.g. the perimeter of a contact hole).

Typically, three sample points are associated with the r^(th) edge. Thefirst sample point is located precisely at the desired target perimeterfor the feature. The intensity at this sample point is denoted as {rightarrow over (s)}·{right arrow over (I)}_(r)(0,z). Thus, x is a localcoordinate that is associated with the edge. Preferably, x is defined insuch a way that x=0 always represents the target edge location. Once aCD tolerance is specified for the feature, other sample points can beestablished. In some cases, particularly with two dimensional patterns,it is not necessary to define both the CD+ and the CD− range limits forthe shape along every cutline. For example, if the pattern consists ofan array of small contact holes, one might place cutlines alonghorizontal and vertical diameters, and also along diagonal diameters.With such a pattern, one would typically specify CD+ and CD− diameterlimits along the horizontal and vertical diameters. Assuming this weredone, it would then not usually be necessary to include the CD− limitsalong the diagonal cutlines, since the four CD− limits along thehorizontal and vertical cutlines would be quite closely positionedwithin the interior of the contact hole. However, in such cases, CD+limits along the diagonals would usually still be necessary.

Placement of the next sample points accounts for exposure conditions.That is, if overexposed, line-like features will contract and space-likefeatures expand. Once a CD tolerance is specified for the feature (or acombined CD+ position shift tolerance), a sample point can be placed ata position along the associated CD slice which would be associated withan extreme position for allowable overexposure. This sample pointposition is denoted as: x=CD+. The intensity at x=CD+, for defocus z, isthen denoted as: {right arrow over (s)}·{right arrow over(I)}_(r)(CD₊,z). Note that overexposure causes edges to be printed inregions where the intensity is lower, so that x=CD+ corresponds to adarker part of the image than does x=0. Likewise, x=CD− designates asample point at the position of worst allowable underexposure, and{right arrow over (s)}·{right arrow over (I)}_(r)(CD⁻,z) represents theintensity at this point. The set of points CD_(r,+) and CD_(r,−) (as ris varied) thus constitute a range of allowable shapes for the maskpatterns.

Exemplary aspects of placement of the additional sample points near keytarget edges are illustrated in FIGS. 2-4. These figures illustrateaspects of the problem of maximizing the integrated common processwindow for the printing of mask patterns as shapes. In the embodimentillustrated, the shapes form lines. The lines are illustrated as darkimages, as in the case of a positive resist. Note that the discussionherein is generally presented in terms of using a positive resist.However, use of a positive resist is not considered limiting, forexample, the method works equally well using a negative resist.

FIGS. 2A-2C illustrate portions of a larger image (not shown). In eachof these figures, a pair of lines 101, 102 is presented. In the largerimage, the line features extend indefinitely in the vertical direction,and are formed by dark regions of the exposing image. Note that suchone-dimensional patterns are only chosen as an example. In general, thepatterns may have an arbitrary two-dimensional shape. In the exemplaryembodiments depicted in FIG. 2, the lines 101, 102 are each 87.5 nmwide. The larger image also contains additional vertical lines that areparallel to the pair shown, and that are spaced apart by the samedistance (i.e., these lines are formed, or “printed”, in periodicarrays). The period is referred to as a “pitch,” and FIG. 2 presentsembodiments over a range of pitches. The exemplary 87.5 nm wide linepatterns repeat periodically, with repeat distance (centerlineseparations) of 245 nm, 275.6 nm, and 315 nm.

Referring to FIG. 2A, shaded region 101 represents a first line 101 ofthe pair of lines. Region 102 represents a second line which isseparated from line 101. The centerline 106, or origin 106, of the firstline 101 is separated from the centerline 107 of the second line 102 by245 nm. The first line 101 has a nominal (design) edge 103 that appearsat a distance that is symmetric about the centerline 106 (i.e., in thiscase, at 43.75 nm). FIG. 2B represents the pair of lines 101, 102 havinga 275.6 nm pitch. FIG. 2C corresponds to the pair of lines 101, 102having a 315 nm pitch.

In the example of FIG. 2, the pitches have a common multiple of 2205 nm,where 245 nm*9=2205 nm; 275.6 nm*8=2205 nm; and, 315 nm*7=2205 nm. Thatis, the three sample pitches illustrated are formed from thosediffraction orders of the 2205 nm common pitch which are multiples of 7,8, or 9. Although the exemplary pitches provided have a common multiplepitch, this is considered illustrative, and not limiting of thisinvention.

One will recognize that achieving exact size or shape may be difficultto realize, and is not even required. That is, for any given shape 101,102 there is generally a range of printed shapes that are acceptable.FIG. 2A depicts aspects of a range of acceptable shapes for theexemplary first line 101.

In FIG. 2A, minimum allowable exposure line 104 is depicted. The minimumallowable exposure line 104 corresponds to an acceptable increase inline width of +9%. This increase is assumed to be the maximum that canbe tolerated in this example. That is, the minimum allowable exposureline 104 is 9% further from the origin 106 than the nominal edge 103.Exposure below a certain dose level that, as explained below, iscalculated from the trial intensities to be the dose that causes theright edge of 101 to be printed at position 104, will cause undesiredundeveloped resist to remain even on the right side of 104. That is,exposure below this dose level will cause the line 101 to print widerthan the allowable +9% tolerance. Therefore, the intensity along theminimum allowable exposure line 104 governs a dose tolerance for theprinted edge. If, for example, the image intensity along the minimumallowable exposure line 104 were 10% larger than the intensity at thenominal edge 103, then a decrease in dose from nominal of more than 10%would cause line feature 101 to print excessively wide.

Similarly, position 105 shows the position of maximum allowableexposure, corresponding to a decrease in line width of 9%. If, forexample, the intensity at position 105 were 10% lower than the intensityalong nominal edge 103, then an increase in exposure beyond 10% ofnominal would cause line feature 101 to print excessively narrow.Exposing patterns in the mask 526 need not take identical shapes to thedesired wafer patterns. For example, FIG. 3 shows a set of shapes foruse in a mask 526 that are designed to print the patterns illustrated inFIG. 2.

Referring to FIG. 3, the background transmission (in regions such as320) of the mask is nominally 100%. Mask line features such as 321 arealmost opaque, and print as line patterns (such as pattern 101) on thewafer. Though dark on the mask, features such as 321 are given aresidual transmission of 6.5% with phase-shift of 180°. This is awell-known method for improving image contrast. The patterns depicted inFIG. 3 for the mask may also employ the well-known assist feature methodfor improving depth of focus. The assist features are shown as 35 nmfeatures such as 323 and 322 that are placed halfway between adjacentline features. The mask features repeat periodically in the horizontaldirection. In FIG. 3, the mask features do not have patterning in thevertical direction. However, it is known that designs with patterningalong a second dimension can, in some cases, give better performanceeven when the desired patterns have a one dimensional appearance.Reference may be had to the publication “Optimum Mask and SourcePatterns to Print a Given Shape,” cited above. It is considered thatvertical patterning would be needed if one wished to print the samepatterns in a 90°-rotated orientation. Although this is not the case forthe example provided, a non-zero periodicity is chosen in the verticaldirection. In fact, a vertical periodicity that is equal to thehorizontal periodicity is used (i.e., a square periodicity is used).Accordingly, each wafer period (245 nm, 275.6 nm, or 315 nm) is depictedin FIG. 3 as one unit in a 3×3 block of repeated unit periods. Thecentral period is indicated in FIG. 3 as a block with different shading,as in block 324. Alternatively, the three unit periods could be regardedas sub-blocks in a common 2205 nm×2205 nm wafer period (including 9×9repetitions of the cell that has 245 nm pitch, or 8×8 repetitions of the275.6 nm pitch cell, or 7×7 repetitions of the 315 nm pitch cell).

The range of doses within acceptable tolerance for a given mask isdetermined by evaluating the image intensity at the extreme boundariesof the allowable range of shapes. For example, and referring back toFIG. 2, the allowable range of doses that will print line feature 101within tolerance is determined by the image intensities along theminimum allowable exposure line 104 and the maximum allowable exposureline 105. In this example, the mask patterns are one-dimensional (FIG.3), so only the intensity at single sample points along lines 104 and105 need be calculated. Sample points are shown in FIG. 4.

FIGS. 4A-4C, collectively referred to as FIG. 4, depict sample points incorrespondence to a shape. In the embodiment shown, the shapecorresponds to the exemplary lines 101, 102 depicted in FIG. 2.

In FIG. 4A, sample points are shown corresponding to the pair of lines101, 102 having a 245 nm pitch depicted in FIG. 2A. A sample point 211correlating to a position along nominal edge 103 is shown. The nominaledge 103 is some specified distance from the centerline 224, in thisexample 43.75 nm, which is half the desired width of line 101. Samplepoints 214 and 215 are shown, and correspond to the single points alongthe minimum allowable exposure line 104 and the maximum allowableexposure line 105, respectively.

Various bright sample points 216 are also shown. Successful printing ofa set of mask patterns requires that each pattern print with appropriatepolarity, as well as with acceptable dimensions. For example, the doseat the bright-polarity sample points 216 must be large enough to ensuredevelopment of resist in this region. Therefore, requirements for thisregion may be established, such as a requirement that the dose be atleast, for example, 120% of a predetermined threshold.

If one desires that the 87.5 nm wide line print with the same CD (inthis example, 87.5 nm) in all pitches using a nominal dose, theintensity at point 211 ({right arrow over (s)}·{right arrow over (I)})is preferably constrained to be about the same as the intensity atsample point 220 and sample point 221, shown in FIG. 4B, and FIG. 4C,respectively. Following the example established in FIG. 2, sample point220 and sample point 221 lie along the nominal edges 103 of the linesfeatures 101 in the 275.6 nm (FIG. 2B) and 315 nm (FIG. 2C) pitchpatterns, respectively.

In an extended region of the image it is not necessary to test theintensity of every bright sample point individually. Instead, the pointscan be grouped in sets, with each set covering image regions of width,for example, ≲0.3λ/NA(1+σ_(Max)), where σ_(Max) refers to the relativedirection cosine (normalized against NA) of the most obliquely incidentillumination direction that can be provided.

A plurality of bright sample points are identified for bright regionsthat are sufficiently extended. When sample points are grouped in sets,it is generally considered adequate to assess image polarity using asparser spacing between the sets than would be needed for ungroupedsample points. The averaged sample points may be treated computationallyas a single point. Therefore, as used herein, the term “sample point”may be used to refer to a group of appropriately averaged sample points.As an example, the 275.6 nm pitch pattern depicted in FIG. 4 includes afirst group of sample points 222, and a second group of sample points223. Note that sample point 223, though explicitly including only thethree points to the left of period centerline 219, can be regarded asaveraging two sets of three points. That is, the sample point 223accounts for the three points to the left of the centerline 219, as wellas the three points symmetrically disposed to the right of centerline219. This follows from the bilateral symmetry of the masks 526 depictedin FIG. 3, and from the bilateral source symmetry discussed below.

If centerline 219 is then taken as the effective position of a samplepoint corresponding to the second group of sample points 223, theseparation between first group of sample points 222 and the second groupof sample points 223 is about 0.37λ/NA(1+σ_(Max)) This separation isconsidered to be a suitable value for spacing bright sample points (andgroups thereof) in extended bright areas. Further, relying on bilateralsymmetry also permits the omission of sample points in half the unitperiod. For example, refer to the area of the chart as shown in FIG. 4B,corresponding to the right half of the line 101.

Preferably, sample points are also added to identify dark regions. Forconvenience, these sample points are referred to as “dark samplepoints.” The intensity at dark sample point 218 may be constrained to beno larger than a selected value. For example, the dark sample point maybe limited to a dose level that is about 40% of a predeterminedthreshold. Note that 218, though a single point, may be used effectivelyas a grouped point since it can be regarded as being symmetricallypaired about the origin with point 217. Sample points may either bespecified manually by the user, or they may be deployed automaticallywith spacings of magnitude of about 0.4λ/NA(1+σ_(Max)), and group widthsof ≦≈0.3λ/NA(1+σ_(Max)).

Source pixelation is now considered for the sample embodiment. As wasdiscussed earlier, the three sample pitches can be associated withparticular diffraction orders from a 2205 nm unit cell (taken to be asquare), namely those orders which are multiples of 7, 8, or 9. Thoughthe example only involves vertical lines, a source pixelation that couldsupport horizontal lines as well may be chosen (though each individualpattern is assumed to be modulated along only one axis, i.e.unstructured along the other).

If a source pixelation method, such as one described in U.S. Pat. No.6,563,566 is applied under these conditions, the (prior art) sourceconstruction shown in FIG. 5 is obtained. Aspects of this method are nowsummarized, and are described in relation to existing (prior art)techniques.

In FIG. 5, outer circle 401 represents the lens pupil 401, of radiusequal to the lens NA. In this example, the numerical aperture (NA)equals 0.75. The two dimensional space of the FIG. 5 graph representsillumination and collection directions. More specifically, the x and yaxes of FIG. 5 represent the x and y direction cosines of light beamsthat illuminate masks (such as mask 526 depicted in FIG. 3), and lightbeams that diffract from them. The inner circle 402 represents themaximum obliquity in illumination that the lithographic tool cansupport. In this case, the radius σ_(Max) of circle 402 (expressed as afraction of the NA, as is customary) is 0.88 NA. The directions at whichdiffraction orders propagate from the mask under illumination at normalincidence are plotted in FIG. 5 as dots, such as dot (order) 403.Additional circles of radius NA, for example circle 404, are centeredabout vertices located at each diffracted order 403. If the mask 526 isilluminated from a direction other than normal incidence, the directionat which each order diffracts will shift, and this shift can cause theorder direction to either shift into or out of collection pupil 401.This is determined by circles like 404. For example, circle 404represents the range of illumination directions which cause order 403 tobe collected by pupil 401. The resulting overlap areas between thecircles, items 1 to 40, represent suitable source pixelations in thecase where only the in-focus image is considered. This is because thesame in-focus image is produced (in the absence of aberrations) by everyillumination direction within a particular overlap region, since eachsuch illuminating direction causes the same set of orders to becollected.

It should be noted, as explained further in U.S. Pat. No. 6,563,566,that the source pixels 1-40 actually represent symmetrical illuminationpatterns, rather than isolated oblique sets of directions. For example,the source region denoted 2 actually refers to the four-fold symmetricillumination pattern that comprises regions 405, 406, and 407, inaddition to the region labeled as 2. Such mirroring is preferred tolimit anomalies (such as image skew) through focus. The intensity ofillumination light for regions 405, 406, and 407 is equal to that forregion 2.

It should also be noted that one does not need to define a commonmultiple pitch when optimizing multiple patterns. An example that obeysthis condition was chosen for FIG. 2 simply to improve clarity in theFIG. 5 pixelation diagram. That is, this example was chosen to providean illustrative case with a visually even and regular appearance.However, in general, pixelation of an illumination source, as introducedin reference to FIG. 5, may be constructed for multiple mask objects bysimply superposing atop one another the construction circles 404 thatarise in the different masks about diffraction orders like 403.

An aspect of this invention is in maximizing the integrated processwindow through focus. It should be noted that a problem with theconstruction shown in FIG. 5 is that a large region, such as region 4,can cover a wide range of illumination directions, whose affect on theimage can vary widely in focal planes other than the plane of bestfocus. Techniques disclosed herein may be used to account for thiseffect by subdividing the larger regions (such as region 4) into smallerunits. One example is shown in FIG. 6. It is considered that it isgenerally reasonable to use subdivisions having an area of about 1/50 or1/100 that of one quadrant of the range of possible illuminatingdirections (i.e., an area that is about 1/200 or 1/400 that of theentire circle 402).

Thus, in FIG. 6 the regions which have a large area, such as regions 4and 24 as shown in FIG. 5, have been subdivided into smaller regions.For example, regions 4 and 24 appearing in FIG. 5 have been divided into12 subdivisions in FIG. 6. Regions depicted in FIG. 5 having small ormoderate area (such as regions 1 or 11) may be retained without furtherdivision. Each of the subdivisions shown in FIG. 6, or the undividedsmall-to-moderate sized regions, represents an independent “sourcepixel.” A source pixel is also referred to as a “source element.”

Referring to FIG. 6, various regions are shown. One may note that asymmetry exists between region 4 and region 24. For convenience, thepixels are identified using two numbers, which correlate to a“region-source element” nomenclature. For example, pixel 1-1 correspondsto the undivided source region 1. Pixel 4-3 corresponds to the sourceelement at position x=0.196, y=0.247, which corresponds to the 3^(rd)subdivision of region 4. As provided for in the construction depicted inFIG. 6, the intensity associated with each source element actuallyrepresents the intensity for a set of mirror-symmetric directionalsource elements. The set preferably contains four members, due to thepreferred four-fold symmetry of the source, i.e. bilateral symmetryabout the horizontal and vertical planes. If one seeks to print the samepatterns in both horizontal and vertical orientations, one can employsets (containing eight members) that are also symmetrical about the45-degree and 135-degree diagonals. Such a set might include, forexample, elements 4-5 and 24-5, shown in FIG. 6.

Once the source elements have been defined, the image intensitiesprovided by a unit source intensity applied to each element arecalculated for each of a number of focal planes. For example, in oneembodiment it is considered reasonable to choose a focal step on theorder of 0.1λ/NA² between the focal planes, and to step through asufficient number of focal planes to ensure that intensities arecalculated for a focal range that will exceed the achievable depth offocus. As an example, in one embodiment, the calculation might bestepped to a maximum defocus of 5λ/NA². The intensity of theelement-illuminated images can be calculated using standard methods, asdiscussed elsewhere herein. FIGS. 7-10 depict examples of intensitycalculated for two elements. These intensities are referred to as trialintensities, as calculation of the trial intensity is a preliminary stepin choosing the intensities that will actually be provided. Theprojections involved need not be carried out in simulation, these mayinstead be determined experimentally.

In FIG. 7, the calculated trial intensity is depicted for source element4-6, when the image is in focus. The graph of FIG. 7 presents curves foreach of the three exemplary embodiments depicted in FIGS. 2-4. FIG. 8depicts the trial intensity produced from source element 4-6, when theimage is defocused, at 0.3 μm. Again, curves for each of the threeexemplary embodiments depicted in FIGS. 2-4 are presented. FIG. 9 andFIG. 10 make a corresponding presentation for source element 3-2.

The image intensity provided by a more complex source can be obtained asa weighted sum of these element-illuminated trial images, since aparticular source element is given a unit intensity when calculating thetrial images (with all other sources given zero intensity). For example,in one embodiment, the image intensity may be obtained by a weighted sumof the corresponding intensity values for each element. Therefore, at agiven sample point (for example, the pth sample point), the trial imageintensities in a particular focal plane (for example, the kth focalplane) constitute a vector of proportionalities {right arrow over(I)}_(p)(kΔz), and the intensity at the sample point from a complexsource defined by a vector of source element intensities {right arrowover (s)} is given by {right arrow over (s)}·{right arrow over(I)}_(p)(kΔz).

The presently preferred methods disclosed herein illuminate the maskwith a set of source elements having optimum intensity {right arrow over(s)} to maximize the integrated process window. In mathematical terms,it is desired to determine the list of source intensities {right arrowover (s)} that solves the following problem:

$\underset{w.r.t.\mspace{14mu}\overset{\rightarrow}{s}}{Maximize}\mspace{14mu}\frac{\begin{matrix}{\int_{0}^{F}{{\mathbb{d}f}\left\{ {{\underset{r}{Min}\left( {\underset{z = 0}{\overset{z = f}{Min}}\left\lbrack \frac{1}{\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{+},z} \right)}} \right\rbrack} \right)} -} \right.}} \\\left. {\underset{r}{Max}\left( {\underset{z = 0}{\overset{z = f}{Max}}\left\lbrack \frac{1}{\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{-},z} \right)}} \right\rbrack} \right)} \right\}\end{matrix}}{\left( \frac{1}{\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{1}\left( {0\text{,}0} \right)}} \right)}$${{{where}\mspace{14mu} F} \equiv {{Min}\left\lbrack {f,{{{such}\mspace{14mu}{that}\mspace{14mu}{\underset{r}{Min}\left( {\overset{z = f}{\underset{z = 0}{Min}}\left\lbrack \frac{1}{\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{+},z} \right)}} \right\rbrack} \right)}} = {\underset{r}{Max}\left( {\underset{z = 0}{\overset{z = f}{Max}}\left\lbrack \frac{1}{\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{-},z} \right)}} \right\rbrack} \right)}}} \right\rbrack}},$subject to constraints:

$\begin{matrix}\begin{matrix}{{\left. A \right)\mspace{14mu}{\sum\limits_{j = 1}^{J_{Max}}\; s_{j}}} \geq S_{Min}} & \; \\{{\left. B \right)\mspace{14mu} s_{j}} \leq S_{{Max},j}} & \left( {{\forall j}❘{1 \leq j \leq J_{Max}}} \right) \\{{\left. C \right)\mspace{14mu} 0} \leq s_{j}} & \left( {{\forall j}❘{1 \leq j \leq J_{Max}}} \right) \\{{\left. D \right)\mspace{14mu}{\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{r}\left( {0\text{,}0} \right)}}} = {{non}\text{-}{present}\mspace{14mu}\text{constant}\mspace{14mu} Q}} & \left( {{\forall r}❘{1 \leq j \leq r_{Max}}} \right) \\{{\left. E \right)\mspace{14mu}{\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{u}\left( {0\text{,}0} \right)}}} \geq {I_{Bright}Q}} & \left( {{\forall u}❘{1 \leq u \leq u_{Max}}} \right) \\{{\left. F \right)\mspace{14mu}{\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{v}\left( {0\text{,}0} \right)}}} \leq {I_{Dark}Q}} & \left( {{\forall v}❘{1 \leq v \leq v_{Max}}} \right)\end{matrix} & \left( {{Eq}.\mspace{14mu}\lbrack 2\rbrack} \right)\end{matrix}$

As can be appreciated, in the above formulation the problem is expressedin a highly nonlinear form. The merit function to be maximized in Eq.[2] (which may also be referred to as an objective function) isessentially the integral of the exposure latitude through the depth offocus. More precisely, the merit function is the integral of thenormalized exposure latitude over all achievable depths of focus. Asmaximizing the percentage (or fractional) exposure latitude across focusis desired, the unknown exposure values appear in the numerator as wellas the denominator of the objective function (Eq. [2]). Note that theabsolute exposure level is not a controlling factor in this context,since it can be adjusted separately on the exposure tool, for example bychanging exposure time. For simplicity, the exposure value chosen forthe denominator in the fractional normalization is taken as the nominalexposure for printing the first (r=1) edge.

Exposure latitude is the range of exposure between the upper and lowerexposure limits, for a given focal range (f). The upper and lowerexposure limits involve double maximizations or double minimizations.For example, the achieved exposure latitude can be no larger than theexposure latitude that is attained by the weakest feature in thepattern. Accordingly, the maximum exposure in the integrand is taken asthe minimum exposure over r of the upper exposure limits attained ateach edge. Likewise the minimum exposure is the maximum over r of thelower exposure limits. In addition, for a particular exposure level tobe considered valid over a given focal range f, it is not sufficientthat the exposure merely print all CDs within tolerance at defocus z=f,the exposure must also be adequate to successfully print the CDs at allintermediate focal planes between z=0 and z=f. Preferably, the variableof integration f is interpreted as focal range and not as a simpledefocus. The maximum allowable exposure in a given focal range f is thusthe minimum of the upper exposure limits achieved across all focalplanes within the range z=0 to z=f.

Similarly, the lower exposure limit is defined in terms of a doublemaximization over r and z. Note that maximizations and minimizations arenonlinear and slope-discontinuous in the variables involved.

The limit of integration F is also defined in a non-linear manner,namely as the focal range at which the exposure latitude first closesdown to zero. Thus, F is the maximum attainable depth-of-focus (DOF),that is, F is the “length” of the ED window if exposure latitude isregarded as the “width” of the window. For simplicity, Eq. [2] neglectsaberrations other than defocus, however this restriction is notnecessary.

It will be clear to those skilled in the art that the constraints in theexemplary problem can be managed in a number of suitable ways.Furthermore, the constraints presented for Eq. [2] are representative,and are not limiting. As presented, constraint A enforces a minimumpupil fill, constraint B sets maximum intensity limits for each sourceelement (e.g., a limit proportional to element area in the pupil),constraint C rules out unphysical negative intensities, constraint Denforces CD requirements on the nominal image, constraint E requiresthat nominally bright regions indeed print bright, and constraint F thatdark regions print dark.

Eq. [4] and Eq. [5] below provide two approximations that can be madefor Eq. [2]. In the first approximation, the integration is replaced bya summation over focus grid points (or planes). These grid points arespaced by a distance Δz which is small compared to the depth of focus.As noted above, a defocus step of Δz=0.1λ/NA² is considered reasonable.

To illustrate the second approximation, a simpler approximation is firstpresented. This is for purposes of illustration only, and is notactually used as the second approximation. First, an assumption is madethat CD errors, and CD tolerances throughout the depth of focus, arenumerically small when compared to the lens resolution. That is, it isassumed that the allowable variation in a feature width is small whencompared to the width of the actual feature. Such an assumption impliesthat the range of intensity variations appearing at image edges will besmall when compared to the baseline intensity level of the nominal imagecontour, as long as the image is printed within the ED window ofinterest. This permits one to make an expansion of the form:

$\begin{matrix}{{{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{\pm},z} \right)} \cong {{{\overset{\rightarrow}{I}}_{1}\left( {0\text{,}0} \right)} + {ɛ_{r} \mp {\frac{\Delta\;{CD}}{2}\frac{\partial{\overset{\rightarrow}{I}}_{r}}{\partial x}}} + {\frac{z^{2}}{2}\frac{\partial^{2}{\overset{\rightarrow}{I}}_{r}}{\partial z^{2}}} + {\frac{z^{4}}{24}\frac{\partial^{4}{\overset{\rightarrow}{I}}_{r}}{\partial z^{4}}} + \ldots}} & \left( {{Eq}.\mspace{14mu}\lbrack 3\rbrack} \right)\end{matrix}$with ε_(r)≡{right arrow over (I)}_(r)(0,0)−{right arrow over (I)}₁(0,0),and where all quantities on the second line are considered smallcompared to the first term, {right arrow over (I)}₁(0,0). Odd powers ofz may be included in Eq. [3], where aberrations are non-zero. Note againthat Eq. [3] is presented for purposes of explanation only.

Assuming the approximation of Eq. [3] is valid, the function in Eq. [2]can be manipulated to obtain the approximation:

$\begin{matrix}{\frac{\begin{matrix}{\int_{0}^{F}{{\mathbb{d}f}\left\{ {{\underset{r}{Min}\left( {\underset{z = 0}{\overset{z = f}{Min}}\left\lbrack \frac{1}{\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{+},z} \right)}} \right\rbrack} \right)} -} \right.}} \\\left. {\underset{r}{Max}\left( {\underset{z = 0}{\overset{z = f}{Max}}\left\lbrack \frac{1}{\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{-},z} \right)}} \right\rbrack} \right)} \right\}\end{matrix}}{\left( \frac{1}{\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{1}\left( {0\text{,}0} \right)}} \right)} \cong \frac{\begin{matrix}{\int_{0}^{F}{{\mathbb{d}f}\left\{ {{\underset{r}{Min}\left( {\underset{z = 0}{\overset{z = f}{Min}}\left\lbrack {\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{-},z} \right)}} \right\rbrack} \right)} -} \right.}} \\\left. {\underset{r}{Max}\left( {\underset{z = 0}{\overset{z = f}{Max}}\left\lbrack {\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{+},z} \right)}} \right\rbrack} \right)} \right\}\end{matrix}}{\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{1}\left( {0\text{,}0} \right)}}} & \left( {{Eq}.\mspace{14mu}\lbrack 4\rbrack} \right)\end{matrix}$

Eq. [4] approximates the percentage (or fractional) of exposure latitudeby fractional intensity latitude. Due to the fractional normalization{right arrow over (s)} continues to appear in both numerator anddenominator, and the integrand continues to be defined by doublemaximizations and double minimizations.

Eq. [4] is the approximation that is actually made in Eq. [2], ratherthan the nominally more restrictive Eq. [3], although Eq. [3] has beenshown to be quite accurate as well. Therefore, the integrated fractionalintensity latitude is maximized through focus, rather than integratedfractional exposure latitude. Note that the integrated fractionalintensity latitude is a reasonable figure of merit, therefore theoptimization disclosed herein need not depend on the Eq. [3]approximation.

As was noted, the right-side integral in Eq. [4] is approximated by adiscrete sum:

$\begin{matrix}{\frac{\begin{matrix}{\int_{0}^{F}{{\mathbb{d}f}\left\{ {{\underset{r}{Min}\left( {\underset{z = 0}{\overset{z = f}{Min}}\left\lbrack {\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{-},z} \right)}} \right\rbrack} \right)} -} \right.}} \\\left. {\underset{r}{Max}\left( {\underset{z = 0}{\overset{z = f}{Max}}\left\lbrack {\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{+},z} \right)}} \right\rbrack} \right)} \right\}\end{matrix}}{\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{1}\left( {0\text{,}0} \right)}} \cong {\frac{\begin{matrix}{\Delta\; z{\sum\limits_{k = 0}^{\frac{k = f}{\Delta\; z}}\;\left\{ {\underset{r}{Min}\left( {\overset{j = k}{\underset{j = 0}{Min}}\left\lbrack {\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{-},{j\;\Delta\; z}} \right)}} \right\rbrack} \right)} \right.}} \\\left. {\underset{r}{Max}\left( {\overset{j = k}{\underset{j = 0}{Max}}\left\lbrack {\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{+},{j\;\Delta\; z}} \right)}} \right\rbrack} \right)} \right\}\end{matrix}}{\overset{\rightarrow}{s} \cdot {{\overset{\rightarrow}{I}}_{1}\left( {0\text{,}0} \right)}}.}} & \left( {{Eq}.\mspace{14mu}\lbrack 5\rbrack} \right)\end{matrix}$

After making the approximations provided in Eq. [4] and Eq. [5], Eq. [2]can be solved by introducing new variables K_(Max), w₁ . . . w_(Kmax),w′₁ . . . w′_(Kmax), making a further change of variables {right arrowover (s)}→

, and then solving the following equations:

${\underset{w.r.t.\mspace{14mu} K_{Max}}{Maximize}\left( {\underset{w.r.t.\mspace{14mu}}{Maximize}\left\lbrack {\Psi\;\left( {K_{{Max};}} \right)} \right\rbrack} \right)},$

-   -   where objective Ψ is defined by:

${{\Psi\left( {K_{Max};} \right)} \equiv {\sum\limits_{k = 1}^{K_{Max}}\;\left( {w_{k}^{\prime} - w_{k}} \right)}},$subject to:

$\begin{matrix}\begin{matrix}{{\left. a \right)\mspace{14mu} 0} \leq {S_{Min}} \leq {S_{{Max},j}{\sum\limits_{j = 1}^{J_{Max}}\;}}} & \left( {{\forall j}❘{1 \leq j \leq J_{Max}}} \right) \\{{\left. b \right)\mspace{14mu}{\cdot {{\overset{\rightarrow}{I}}_{r}\left( {0\text{,}0} \right)}}} = 1} & \left( {{\forall r}❘{1 \leq r \leq r_{Max}}} \right) \\{{\left. c \right)\mspace{14mu}{\cdot {{\overset{\rightarrow}{I}}_{u}\left( {0\text{,}0} \right)}}} \geq I_{Bright}} & \left( {{\forall u}❘{1 \leq u \leq u_{Max}}} \right) \\{{\left. d \right)\mspace{14mu}{\cdot {{\overset{\rightarrow}{I}}_{v}\left( {0\text{,}0} \right)}}} \leq I_{Dark}} & \left( {{\forall v}❘{1 \leq v \leq v_{Max}}} \right) \\{{\left. e \right)\mspace{14mu} w_{k}} \geq {\cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{+},{k\;\Delta\; z}} \right)}}} & \left( {{\forall r},{k❘{1 \leq r \leq r_{Max}}},} \right. \\\; & \left. {0 \leq k \leq K_{Max}} \right) \\{{\left. f \right)\mspace{14mu}{w_{0}^{\prime}}_{\;}} \geq {\cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{+},{k\;\Delta\; z}} \right)}}} & \left( {{\forall r},{k❘{1 \leq r \leq r_{Max}}},} \right. \\\; & \left. {0 \leq k \leq K_{Max}} \right) \\{{{\left. g \right)\mspace{14mu}{\cdot {{\overset{\rightarrow}{I}}_{1}\left( {0\text{,}0} \right)}}} \geq {\cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{+},0} \right)}}}\mspace{14mu}} & \left( {{\forall r}❘{1 \leq r \leq r_{Max}}} \right) \\{{{\left. h \right)\mspace{14mu}{\cdot {{\overset{\rightarrow}{I}}_{1}\left( {0\text{,}0} \right)}}} \leq {\cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{-},0} \right)}}}\;} & \left( {{\forall r}❘{1 \leq r \leq r_{Max}}} \right) \\{{\left. i \right)\mspace{14mu}{w_{k}^{\prime}}_{\;}} \leq {\cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{-},{k\;\Delta\; z}} \right)}}} & \left( {{\forall r},{k❘{1 \leq r \leq r_{Max}}},} \right. \\\; & \left. {0 \leq k \leq K_{Max}} \right) \\{{\left. j \right)\mspace{14mu}{w_{0}}_{\;}} \leq {\cdot {{\overset{\rightarrow}{I}}_{r}\left( {{CD}_{-},{k\;\Delta\; z}} \right)}}} & \left( {{\forall r},{k❘{1 \leq r \leq r_{Max}}},} \right. \\\; & \left. {0 \leq k \leq K_{Max}} \right) \\{{\left. k \right)\mspace{14mu} w_{k}} \geq w_{k - 1}} & \left( {{\forall k}❘{1 \leq k \leq K_{Max}}} \right) \\{{\left. l \right)\mspace{14mu} w_{k}^{\prime}} \geq w_{k - 1}^{\prime}} & \left( {{\forall k}❘{1 \leq k \leq K_{Max}}} \right) \\{{\left. m \right)\mspace{14mu} w_{k}} \leq w_{k}^{\prime}} & \left( {{\forall k}❘{0 \leq k \leq K_{Max}}} \right)\end{matrix} & \left( {{Eq}.\mspace{14mu}\lbrack 6\rbrack} \right)\end{matrix}$

The required source intensities are then obtained from:

s j = ⁢ Min j ⁡ [ S Max , j ] . ( Eq . ⁢ 7 )

Before explaining the derivation of Eq. [6] and Eq. [7], it isinstructive to first consider their general structure. If a fixed valueis chosen for K_(Max), Eq. [6] becomes a pure linear programming problemin

, which is easily solved as linear programming problems can be solvedglobally using known algorithms that are highly efficient. Theconstraints include some redundancy (for example, constraint j followsfrom constraints i and m). Depending on the linear program solver used,one may choose to eliminate redundant constraints like j. The outermaximization for K_(Max) remains to be solved and is highly nonlinear;however, it only involves a single variable. Therefore it is feasible tofind the global solution for K_(Max) using a simple one dimensional gridsearch. This type of a search may be referred to herein as a “focalrange loop.” Moreover, for all cases tested, it was found that the outerK_(Max) dependence exhibited a simple single-peaked behavior. Thus, itis possible to optimize K_(Max) by local search. For example, one maystart at a conservative value of K_(Max) given by the ratio of theconventional solution DOF to Δz, and one finds that as K_(Max) isincreased, the objective function first increases steadily to a maximumand then rapidly decreases upon further expansion of the depth of focus,and quickly reaches a point where the constraints are no longerachievable. When aberrations are small but nonzero, one may also solvefor the lower focal limit by making a local search about z=−Δz K_(Max)by testing successively z=Δz K_(Max), z=−Δz (K_(Max)−1), z=−ΔZ(K_(Max)+1), etc.

To explain how Eq. [6] and Eq. [7] may be used to solve Eq. [2], Eq.[4], and Eq. [5], it is first noted that the structure of the left-sideof Eq. [4], while strongly nonlinear in numerical terms, has analgebraic structure that is predominantly linear in {right arrow over(s)}, and that can be converted into a linear equivalent.

An immediate, but seemingly improper way to remove the fractionalnonlinearity (i.e., the nonlinearity that arises because {right arrowover (s)} appears in both numerator and denominator), is to simplyconstrain the denominator to be 1 (i.e., the focused intensity {rightarrow over (s)}·{right arrow over (I)}₁(0,0) at the first edge positionis constrained to have a value of unity). The remaining task for solvingEq. [4] is then determining a maximum value for the numerator alone,which is closer to solving a linear problem. However, it is noted thatimposing an artificial requirement of unit intensity at edge position 1will almost always force a non-optimal solution, so that in itself sucha unit-intensity requirement does not provide the desired solution.While it is sometimes true that a fully optimal solution can be rescaledwithin a limited range (e.g., to adjust edge intensity at position 1 toa specific value such as 1), achievement of unit intensity at edgeposition 1 would generally require that either constraint A orconstraint B for Eq. [2] be violated. Resealing the optimal solutionmeans increasing or decreasing all source intensities by a commonfactor. However, it is considered that this would eventually (usuallyrather quickly) either cause the overall source intensity to drop belowS_(Min) (constraint A), or the intensity of a particular sourceintensity (e.g., the j^(th)) to increase above S_(max,)j (constraint B).

However, without unrecoverable loss of generality, it is considered thatthe problem can be temporarily reformulated so that constraints A and Bare replaced by a single constraint. This constraint requires that thefraction of the total source intensity of a given source element be nolarger than ratio of the maximal allowable intensity of that element tothe total required source intensity. The source intensities satisfyingthis revised constraint will not in general satisfy the constraints inEq. [2], so these changed source intensities are denoted with the newsymbol

, rather than the symbol {right arrow over (s)} used previously. Inmathematical terms, the new merged version of constraints A and B isthen:

∑ j = 1 J Max ⁢ ≤ S Max , j S Min Eq . ⁢ [ 8 ]which, when cast in linear form, yields constraint a for Eq. [6]. Notethat as written, constraint a also includes constraint C of Eq. [2].Expressed in the

 variables, constraint b of Eq. [6] provides a linearizing constraintthat the focused intensity at the first edge position have a value ofunity.

Once Eq. [6] is solved for

, the desired global solution to Eq. [2] (as modified by Eq. [4] and Eq.[5]) can be determined by using Eq. [7]. This global solution satisfiesthe original constraints A and B. Note that the intensity at edgeposition 1 is no longer artificially set to unity in the final solutionfor {right arrow over (s)}, which is as it should be since theunit-intensity constraint was merely adopted as an intermediate aid tosolution.

Even after the above changes, the numerator of Eq. [5] still containsnon-linear characteristics arising from the maximizations andminimizations, and from the presence of the maximum focal window extentF (defined in Eq. [2]). As noted above, the non-linearity in F can beaccounted for by simply stepping through possible terminating focalplanes in an outer solution loop. This process assures globalapplicability of the solution. However, in practice, single-parameterlocal optimization in the outer loop can be used to reliably solve forF.

Next, the nonlinear characteristics represented by the embeddedminimizations and maximizations in Eq. [2] and Eq. [5] are considered.These are removed by reformulating the objective for Eq. [5] in terms ofadditional variables that are subject to linear constraints.Specifically, new variables w′ and w are introduced into each of thestepped focal planes along the z-axis, with w′ essentially representingthe darkest of the bright boundaries of the allowable range of shapes,and w representing the brightest of the dark boundaries. The linearconstraints that introduce the w and w′ variables, namely constraints iand e in Eq. [6], do not define these variables so stringently; instead,they merely impose the weaker requirements that w′_(k) (and w_(k)) bebelow (or above) the intensities at all bright-boundary (ordark-boundary) sample points in the k^(th) focal plane. The w′_(k) andw_(k) variables are actually pushed to those strict limits duringmaximization of the objective function Ψ in Eq. [6]. This isautomatically achieved in our formulation because any withdrawal ofthese variables from the i and e binding limits would degrade theobjective function Ψ without easing any other constraints. Hence suchwithdrawal cannot occur when the objective function Ψ is optimally setby the linear program solver.

Constraint k in Eq. [6] ensures that, in addition to being brighter thanall dark-boundaries in the k^(th) focal plane, w_(k) is also brighterthan the dark-boundaries in all focal planes within the truncated range0≦f≦kΔz. In fact, w_(k) achieves this brightest value duringoptimization of Ψ, for the reason explained in the previous paragraph.Thus, 1/w_(k) represents the largest exposure that keeps the pattern intolerance throughout the focal range f≦kΔz, while 1/w′_(k) representsthe smallest allowable exposure. Since w_(k) and w′_(k) take on thesevalues, the Eq. [6] objective function Ψ(

)=Σ(w′_(k)−w_(k)) represents, when maximized, the integrated processwindow (within the accuracy of the Eq. [4] and Eq. [5] approximations).As previously explained, use of the new variable

, instead of {right arrow over (s)}, insures that it is the fractionalexposure latitude that Ψ integrates.

FIG. 11 and FIG. 12 explain in flowchart form the above procedure forobtaining the source solution. FIG. 11 shows the fall procedure, andFIG. 12 shows within the dashed box an expanded description of the focalrange loop.

A summary of the preferred embodiment for obtaining the source solutionis provided in FIG. 11. In step 1101, a user supplies mask shapes and anallowable range of printed shapes, and optionally a source pixelation.In step 1102, sample points are determined. In step 1103, sourcepixelation is determined (if not performed in step 1101). In step 1104,the user determines proportionalities between the source intensities andthe sample point intensities, for multiple focal planes. In step 1105,maximum allowable intensity at dark sample points, and minimum allowableintensity at bright sample points is determined. In step 1106, anintensity parameter is constrained in each focal plane to the maximumintensity among sample points at dark boundaries of the range for theshape. In step 1107, an intensity parameter in each focal plane isconstrained to the minimum intensity among sample points at brightboundaries of the range for the shape. In step 1108, an intensityparameter in each focal plane is constrained to the maximum intensitybound within a truncated local range. In step 1109, an intensityparameter in each focal plane is constrained to the minimum intensitybound within the truncated focal range. In step 1110, a focal range loopis entered.

FIG. 12 illustrates a preferred embodiment of the focal range loop (step1110 of FIG. 11). In FIG. 12, an initial defocus limit is chosen in step1201. Subsequently, the difference between the sum of bounded maximumintensities and the sum of the bounded minimum intensities is maximizedin step 1202. A determination is then made whether a first or seconditeration through the focal range loop is underway, in step 1203. Ifthis is a first iteration through the loop, then a determination is madein step 1204 whether constraints were met. If the constraints were met,then the defocus limit is increased in step 1205. The actions in step1202 are again undertaken, and a determination is again made in step1203 as to whether this is a first or second iteration. The defocuslimit is increased in step 1206 if constraints were met and theobjective improved in the second iteration, otherwise the defocus limitis decreased. The focal range loop terminates in step 1207 with outputof results that provide the maximum sum of differences between theminimum and maximum intensities.

FIG. 13 shows the source solution obtained by this procedure in theexample of the FIG. 2 patterns. In tabular form, the FIG. 13 solutionis:

Source Region Source Intensity  4-9 0.017601  4-11 0.017764 23-10.023122 24-4 0.001306 24-5 0.017662 24-6 0.012216 29-1 0.010328

FIG. 14 depicts one embodiment of a system 540 that includes a systemdriver 528 for operating in accordance with the teachings herein.Preferably, the system driver 528 communicates with, and may control,the illumination controller 522. Preferably, the system driver 528provides for execution of instructions for obtaining the sourcesolution. The system driver 528 may include any equipment as appropriatefor execution of the instructions. Non-limiting examples of systemdriver 528 components include a computer, software (a computer program),a computer readable medium or memory (such as a magnetic and/or opticaldisk drive, tape, semiconductor storage, and other types of memory).Other system driver components may be included. Preferably, the system540 makes use of other components present in prior art lithographysystems 520, such as those shown in FIG. 1.

OTHER EMBODIMENTS

Many modifications to the above method are possible, and several(non-limiting) examples are now provided.

For example, the method can be extended to handle multiple exposures.The different exposures can be carried out with different masks, opticalparameters, polarizations, pupil filters, or focal settings.

Any exposure model that is linear in the source intensity can beaccommodated (so long as the intensity at a point can be expressed inthe form {right arrow over (s)}·{right arrow over (I)}). This caninclude, for example, the effect of resist blur.

The process window can be optimized to reduce the effect of typical maskerrors. For example, instead of optimizing the source to maximize thecommon window for printing several image patterns each with its own mask(e.g., maximizing common window for printing the patterns of FIG. 2using the masks of FIG. 3), the common window for printing each singleimage feature using a variety of different masks can be maximized, wherethese masks may differ from one another only in being perturbed bydifferent simulated fabrication errors. For example, the common windowof a line feature 101 might be optimized over two mask patterns, eachcontaining a line 101 that is rendered on the mask with either thelargest or smallest width that the mask supplier would permit whentrying to achieve nominal mask width. In this way an ideal balance canbe struck in a tradeoff that may arise between optimizing the error-freeimage and minimizing the sensitivity to mask error.

Constraints can be added to enforce minimum acceptable exposure latitudein the image, either at best focus, or in a particular defocused plane.

It is not necessary to constrain the image to exactly achieve thenominal CD at best focus and exposure across all sample cross sections.Such equality constraints can instead be replaced by band constraints.Further, one can choose to remove these constraints altogether, since intheir absence the ED objective function continues to constrain all CDsto at least remain within tolerance, which in some cases may be all thatis desired.

Two-pass strategies can be devised that make use of less rigorous CDconstraints along features such as lines 101, described above. Consider,for example, the following procedure: An initial source optimization iscarried out with no CD equality constraints and with relaxed CDtolerances. For example, one might impose CD tolerances for the EDwindow that are two times less restrictive than the tolerances actuallydesired. Next, with the source held fixed at the newly found solution,an optical proximity correction (OPC) program is used to restore the CDsto their nominal values at best focus and exposure. The source can thenbe re-optimized for the modified mask with the full CD constraints andtolerances in place. One can be sure of convergence as long as the OPCprogram provides guaranteed local convergence.

Constraints on the source intensities can be added that enforce aresemblance of the source to some nominal illumination pattern. Thisensures that image quality in non-critical features will not be severelyimpacted when the optimization is carried out using only a few criticalfeatures.

So far, a method for obtaining an optimum process latitude as predictedby a model has been described. That is, any model in which the intensityat a given point can be expressed as {right arrow over (s)}·{right arrowover (I)}. A similar approach can be used to provide an empirical yieldimprovement in a manufacturing context. The general idea is to limit theallowable changes in the source intensities while deliberatelyspecifying target CDs that do not match a design. This allows anempirical counter-bias to be introduced for, as an example, correctingsmall measured errors in printed features. If the exposure tool has aprogrammable illuminator, this method would allow such targetedcorrections to be carried out very quickly, without ordering a new mask.

Some of the foregoing modifications are required to prevent excessdegradation of non-critical features when source optimization is appliedto critical features. This is realized by limiting the departure of theoptimized source from some nominal illumination pattern. Further,constraints may be adopted that set upper limits to the intensity ofparticular source elements that have been found (in previouscalculations) to adversely affect the process window for patterns otherthan the current critical features being used as targets.

One skilled in the art will recognize that the invention disclosedherein is not limited to printing a mask with maximum possible processwindow through adjustment of the source distribution, as disclosedherein, and that the teachings herein may be employed in a variety ofembodiments. As examples, other embodiments are disclosed herein forenhancing operation of a lithography system and the resulting featuresproduced. Accordingly, it is considered that these and other additionalembodiments are within the teachings of this invention, which isdescribed by the appended claims.

1. A method to obtain an illumination source solution for illuminatingat least one mask for printing a pattern defined by the at least onemask, comprising: supplying mask shapes and an allowable range ofprinted shapes; determining sample points; using a source pixelation,determining proportionalities between source intensities and samplepoint intensities, for individual ones of a plurality of focal planes;determining a maximum allowable intensity at dark sample points, and aminimum allowable intensity at bright sample points; constraining afirst intensity parameter in each focal plane to a maximum intensityamong sample points at dark boundaries of the shape range; constraininga second intensity parameter in each focal plane to a minimum intensityamong sample points at bright boundaries of the shape range;constraining the first intensity parameter in each focal plane to amaximum intensity bound within a truncated focal range; constraining thesecond intensity parameter in each focal plane to a minimum intensitybound within the truncated focal range; and executing a focal rangeloop, comprising selecting an initial defocus limit; maximizing adifference between a sum of the bounded maximum intensities and a sum ofthe bounded minimum intensities; and iterating by increasing the defocuslimit so long as the constraints are met, otherwise terminating thefocal range loop and outputting a result that provides a maximumdifference between the sum or the maximum intensities and the sum of theminimum intensities.
 2. A computer program stored on a computer readablemedia, comprising program code to obtain an illumination source solutionfor illuminating at least one mask for printing a pattern defined by theat least one mask, comprising a program code segment, responsive topredetermined mask shapes, an allowable range of printed shapes and anillumination source pixelation, to determine sample points; to determineproportionalities between source intensities and sample pointintensities for individual ones of a plurality of focal planes; todetermine a maximum allowable intensity at dark sample points, and aminimum allowable intensity at bright sample points; to constrain afirst intensity parameter in each focal plane to a maximum intensityamong sample points at dark boundaries of the shape range; to constraina second intensity parameter in each focal plane to a minimum intensityamong sample points at bright boundaries of the shape range; toconstrain a third intensity parameter in each focal plane to a maximumintensity bound within a truncated focal range; to constrain a fourthintensity parameter in each focal plane to a minimum intensity boundwithin the truncated focal range; and to execute a focal range loop toobtain a defocus limit that represents a maximum difference between theminimum and maximum intensities.
 3. A computer program as in claim 2,where the computer program code for the focal range loop comprisesprogram code to select an initial defocus limit; to maximize adifference between a sum of the bounded maximum intensities and a sum ofthe bounded minimum intensities; and to iterate by increasing thedefocus limit so long as the constraints are met, and to otherwiseterminate the focal range loop and output a result that provides amaximum difference between the minimum and maximum intensities.